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Journal of Mathematical Physics

The propagator of the time-dependent forced harmonic oscillator with time-dependent damping

Bin Kang Cheng

Published Jan 1, 1986• 11 citations
Publisher

Abstract

By using the Caldirola–Kanai Hamiltonian and by applying Feynman polygonal paths, exact evaluation of the propagator of one-dimensional time-dependent forced and damped harmonic oscillator, based on the work of Montroll in deriving the propagator of the harmonic oscillator with time-dependent frequency, is possible. The results are compared with previously found results.

DOI
10.1063/1.527364

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•Jan 1, 1983•1 citation
By using the Caldirola–Kanai Hamiltonian and by applying Feynman polygonal paths, exact evaluation of the propagator of one-dimensional time-dependent forced and damped harmonic oscillator, based on the work of Montroll in deriving the propagator of the harmonic oscillator with time-dependent frequency, is possible. The results are compared with previously found results.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.